Solve $\log _9 5=7-3 X$ By Graphing. What Equation(s) Should Be Graphed?A. $y_1=\frac{\log 5}{\log 9}$B. $y_1=\frac{\log 9}{\log 5}$C. $y_2=7-3 X$D. $y_2=-3 X$

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Introduction

Logarithmic equations can be challenging to solve, especially when they involve variables in the exponent. In this article, we will explore how to solve a logarithmic equation using graphing. We will use the equation log⁡95=7−3x\log _9 5=7-3 x as an example and determine which equation(s) should be graphed to solve it.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithm. The logarithm of a number is the exponent to which a base number must be raised to produce that number. For example, log⁡95\log _9 5 is the exponent to which 9 must be raised to produce 5.

The Given Equation

The given equation is log⁡95=7−3x\log _9 5=7-3 x. This equation involves a logarithm with base 9 and a variable xx in the exponent.

Graphing the Equation

To solve the equation using graphing, we need to graph two equations:

  1. The equation log⁡95=7−3x\log _9 5=7-3 x in terms of yy.
  2. The equation y=7−3xy=7-3 x.

Graphing the Logarithmic Equation

To graph the logarithmic equation, we need to rewrite it in terms of yy. We can do this by using the change-of-base formula:

log⁡95=log⁡5log⁡9\log _9 5=\frac{\log 5}{\log 9}

So, the equation becomes:

y=log⁡5log⁡9y=\frac{\log 5}{\log 9}

This is the equation that we will graph first.

Graphing the Linear Equation

The second equation is a linear equation:

y=7−3xy=7-3 x

This equation represents a straight line with a slope of -3 and a y-intercept of 7.

Graphing the Equations

To graph the equations, we can use a graphing calculator or a computer program. We will graph both equations on the same coordinate plane.

Graphing the Logarithmic Equation

When we graph the logarithmic equation y=log⁡5log⁡9y=\frac{\log 5}{\log 9}, we get a horizontal line at y=log⁡5log⁡9y=\frac{\log 5}{\log 9}.

Graphing the Linear Equation

When we graph the linear equation y=7−3xy=7-3 x, we get a straight line with a slope of -3 and a y-intercept of 7.

Finding the Intersection Point

To solve the equation, we need to find the intersection point of the two graphs. The intersection point is the point where the two graphs meet.

Finding the Intersection Point

To find the intersection point, we can set the two equations equal to each other:

log⁡5log⁡9=7−3x\frac{\log 5}{\log 9}=7-3 x

We can then solve for xx:

x=7−log⁡5log⁡93x=\frac{7-\frac{\log 5}{\log 9}}{3}

This is the value of xx that satisfies the equation.

Conclusion

In this article, we have shown how to solve a logarithmic equation using graphing. We have graphed two equations: the logarithmic equation y=log⁡5log⁡9y=\frac{\log 5}{\log 9} and the linear equation y=7−3xy=7-3 x. We have then found the intersection point of the two graphs, which is the solution to the equation.

The Final Answer

The final answer is:

  • A. y1=log⁡5log⁡9y_1=\frac{\log 5}{\log 9}

Introduction

In our previous article, we explored how to solve a logarithmic equation using graphing. We graphed two equations: the logarithmic equation y=log⁡5log⁡9y=\frac{\log 5}{\log 9} and the linear equation y=7−3xy=7-3 x. We then found the intersection point of the two graphs, which is the solution to the equation.

Q&A

Q: What is the purpose of graphing the logarithmic equation? A: The purpose of graphing the logarithmic equation is to find the value of the logarithm. In this case, we graphed the equation y=log⁡5log⁡9y=\frac{\log 5}{\log 9} to find the value of log⁡95\log _9 5.

Q: Why do we need to graph the linear equation? A: We need to graph the linear equation y=7−3xy=7-3 x to find the value of xx that satisfies the equation. The linear equation represents a straight line with a slope of -3 and a y-intercept of 7.

Q: How do we find the intersection point of the two graphs? A: To find the intersection point of the two graphs, we set the two equations equal to each other and solve for xx. In this case, we set the equation y=log⁡5log⁡9y=\frac{\log 5}{\log 9} equal to the equation y=7−3xy=7-3 x and solved for xx.

Q: What is the value of xx that satisfies the equation? A: The value of xx that satisfies the equation is x=7−log⁡5log⁡93x=\frac{7-\frac{\log 5}{\log 9}}{3}.

Q: Can we use graphing to solve any logarithmic equation? A: Yes, we can use graphing to solve any logarithmic equation. However, we need to make sure that the equation is in the form y=log⁡aby=\log _a b or y=axy=a^x, where aa is the base of the logarithm and bb is the argument of the logarithm.

Q: What are some common mistakes to avoid when graphing logarithmic equations? A: Some common mistakes to avoid when graphing logarithmic equations include:

  • Not using a graphing calculator or computer program to graph the equations.
  • Not setting the two equations equal to each other to find the intersection point.
  • Not solving for xx correctly.
  • Not checking the solution to make sure it satisfies the original equation.

Q: Can we use graphing to solve systems of logarithmic equations? A: Yes, we can use graphing to solve systems of logarithmic equations. However, we need to make sure that the equations are in the form y=log⁡aby=\log _a b or y=axy=a^x, where aa is the base of the logarithm and bb is the argument of the logarithm.

Conclusion

In this article, we have answered some common questions about solving logarithmic equations using graphing. We have discussed the purpose of graphing the logarithmic equation, why we need to graph the linear equation, and how to find the intersection point of the two graphs. We have also discussed some common mistakes to avoid when graphing logarithmic equations and how to use graphing to solve systems of logarithmic equations.

The Final Answer

The final answer is:

  • A. y1=log⁡5log⁡9y_1=\frac{\log 5}{\log 9}

This is the equation that we should graph to solve the equation log⁡95=7−3x\log _9 5=7-3 x.